Abstract— Constant power loads (CPLs) are often the cause of

instability and no equilibrium of DC microgrids. In this study, we

analyze the existence and stability of equilibrium of DC

microgirds with CPLs and the sufficient conditions for them are

provided. To derive the existence of system equilibrium, we

transform the problem of quadratic equation solvability into the

existence of a fixed point for an increasing fractional mapping.

Then, the sufficient condition based on the Tarski fixed-point

theorem is derived. It is less conservative comparing with the

existing results. Moreover, we adopt the small-signal model to

predict the system qualitative behavior around equilibrium. The

stability conditions are determined by analyzing quadratic

eigenvalue. Overall, the obtained conditions provide the

references for building reliable DC microgrids. The simulation

results verify the correctness of the proposed conditions.

Index Terms-- DC microgrid, solvability, nonlinear equations,

fixed-point theorem, quadratic eigenvalue problem, stability,

constant power load. I. INTRODUCTION

ECENTLY, DC microgrids have attracted much attention

given three main advantages over their AC counterparts,

namely high efficiency, simple control and robustness. They

are increasingly used in applications such as aircraft,

spacecraft and electric vehicle [1]–[3]. In DC microgrid, most

loads are connected to the DC-bus through power electronic

interface circuits which make the loads behave constant power

loads (CPLs) [4].

Nevertheless, the negative impedance of CPLs often result

in instability. Thus, stability has been widely investigated in

DC microgrids with CPLs, and several stabilization methods

have been proposed [5]–[20]. The topologies of DC

microgrids in these studies can be divided into two groups

regarding the number of converters and loads: 1) n converters

and one CPL; 2) n converters and m CPLs.

Various studies focused on stability criteria and stabilization

methods of DC microgrids composed of n converters and one

CPL [5]-[19]. In this topology, the DGs are connected to the

CPLs through a DC bus, and when the DC-bus resistance is

neglected, the loads can be modeled as a single CPL.

Therefore, this system is equivalent to star-connection

topology consisting of n DGs and one CPL. The existence of

equilibrium in this system is determined by a quadratic

equation with one unknown, and hence the sufficient

conditions can be easily obtained [12], [19]. Therefore, related

studies mainly focus on overcoming the instability due to CPL.

Some linear techniques are used to stabilize the system [5]–[8].

Based on the idea that increasing damping can mitigate

oscillations, several stabilization methods such as passivity

based control [5], active damping [6], and virtual impedance

[7]–[8] have been proposed. Furthermore, several nonlinear

methods such as phase-plane analysis [9], feedback

linearization [10] and sliding-mode control [11] have been

applied to overcome CPL instability. To estimate the region of

attraction around known equilibria, Lyapunov-like functions

have been proposed, including Lure Lyapunov function [12],

Brayton-Moser’s mixed potential [13]–[14], block

diagonalized quadratic Lyapunov function [15] and Popov

criterion [16]. In addition, the stability of DC microgrids

under droop control was analyzed in [17]–[19]. A stability

condition is derived based on the reduced-order model of DC

microgrids [17]. Another stability condition is derived under

the assumption that all the DG filter inductances have the

same ratio, R/L [18]. To obtain more accurate stability

conditions, a high dimensional model is used in stability

analysis [19].

Due to the transmission loss, the increase of CPL may result

in the loss of system equilibrium (i.e., voltage collapse) [20]

[22]. Hence, finding the condition for the existence of the

equilibrium is a prerequisite. The condition for equilibrium

existence can be easily obtained under the assumption that the

DC-bus resistance can be neglected. However, in most cases,

this assumption is not reasonable. In the practical mesh

DC-microgrid consisting of n converters and m CPLs, the

existence of equilibrium is determined by an m-dimensional

quadratic equation with m unknowns, which is a difficult

problem. By using quadratic mapping, a necessary condition

for the existence of equilibrium based on a linear matrix

inequality (LMI) is obtained in [20], where the condition is

sufficient if and only if the quadratic mapping is convex.

Although a test to verify the convexity of quadratic mappings

is given in [21], the mappings are usually nonconvex for the

case m > 2. To determine the sufficient condition for the

solvability of the m-dimensional quadratic equation, a

contraction mapping is constructed ingeniously in [22].

However, the obtained condition is a little conservative

because contraction mapping requires that the norm of the

mapping Jacbian matrix is less than 1.

In this study, we investigate the following two questions.

1) Under what conditions should the system admit a constant

steady state and how can we reduce the conservativeness?

2) How to design a stabilization method be designed to

guarantee system stability?

The main contributions of this paper can be summarized as

A control method based on virtual resistance and virtual

inductance concept to overcome instability is proposed.

The sufficient condition for the system to admit an

equilibrium is obtained, and it is less conservative

compared with [22].

Existence and Stability of Equilibrium of DC

Microgrid with Constant Power Loads Zhangjie Liu, Mei Su, Yao Sun, Wenbin Yuan and Hua Han

R

The equilibrium stability is analyzed and the analytical

stability conditions are determined using eigenvalue

analysis.

The rest of this paper is organized as follows: Section II

introduces some preliminaries and notations. Section III

describes the basic models and the proposed control scheme.

The existence of equilibrium for DC microgrids is presented

in Section IV. The stability analysis and the sufficient

conditions are detailed in Section V. Simulation results are

presented in section VI. Finally, we draw our conclusions in

Section VII.

II. PRELIMINARIES AND NOTATION

Definition 1. We denote A > 0 if matrix A is positive definite.

Matrix A (or a vector) is called nonnegative (respectively

positive, negative and nonpositive) if its entries are

nonnegative (respectively positive, negative and nonpositive)

and we denote , , ,A B A B A B A B if the entries of A–B

are all positive, nonnegative, negative and nonpositive,

respectively. In addition, 1n (0n) is the vector of ones (zeros).

For Hermitian matrix A∈ m mR , we denote its eigenvalue as

1 mA A .

Definition 2. Square matrix A is a Z-matrix if all the

off-diagonal elements are zero or negative, and it is also an

M-matrix [23] if and only if one of these statements is true:

1) the eigenvalues of A are in the right half-plane;

2) there exists a positive vector, x, such that 0nAx .

Definition 3. Matrix A∈ n nR is irreducible if there exists no

permutation matrix P such that PTAP can be represented as

11 12

22

TA A

P APO A

where A11, and A22 are square matrices, and O is the zero

matrix of proper dimension [24].

Lemma 1. If A is an M-matrix, then

1) A + D is an M-matrix for every nonnegative diagonal

matrix D;

2) if A is irreducible, A-1 is positive [25].

Lemma 2. Tarski fixed-point theorem [26]. Given D n nR convex, let :f D D be a continuous function such

that

1) f(x) is strictly increasing, i.e., 1 2, Dx x , if 1 2x x ,

1 2f x f x ;

2) 1 2,x x D such that 11 22x xf x f x ,

there is a unique vector 2*

1x xx such that f (x*) = x*.

Lemma 3. Let A be a real positive matrix. Perron root χ and

Perron vector η satisfy Aη = χη, where η 0 and ηTη = 1.

Moreover, χ is also the spectral radius of A, denoted ρ(A).

If A B O , then ρ(A)> ρ(B) [24].

Lemma 4. Let2( )Q M K C . Let M, K and C be

positive definite Hermitian matrices. For a quadratic

eigenvalue problem2 0M K C , Re (λ) <0 [27].

III. DC-MICROGRID MODEL AND CONTROL SCHEME

A general DC microgrid with n converters (DGs) and m

CPLs is illustrated in Fig. 1 and consists of three main

components: sources, loads and cables. In a low-voltage DC

microgrid, the cable can be regarded as purely resistive and we

assume the loads as CPLs. In addition, we consider the

DC-microgrid topology as a graph with the sources and loads

representing nodes, and the cables representing edges,

respectively. Furthermore, we assume the graph as being

strongly connected, i.e., every source has access to every load.

10P

4,10r

Topology in the graph

DG1

DG4

DG2

DG3

7P

9P

6P

5P

8P6,9r

6,7r

DC/DC converter

iV

iL

iC

ijy

id

Fig.1. Diagram of the DC microgrid.

The dynamics of the ith converter can be described by

i

i

L

i i i i

ii L i

diL V d u

dt

duC i i

dt

(1)

where Vi, di, iLi , ui and ii are the input voltage, duty cycle,

inductance current, output voltage and current of the converter,

respectively.

To prevent the instability caused by CPLs in a DC

microgrid, we propose a controller based on virtual resistance

and virtual inductance whose duty cycle is given by

ref 1,2, ,i

i ii i L i

i i i

L ud u k i u i n

V X V , (2)

where Xi, ki, and uref are the virtual inductance, virtual

resistance, and reference voltage of the ith converter,

respectively. Then, substituting (2) into (1) and representing

the result in matrix form, we obtain

LL S

SL S

diX V Ki u

dt

duC i i

dt

(3)

where 1 2 n

T

L L L Li i i i , uS= [u1 u2 … un]T, iS = [i1 i2 …

in]T, V*= uref1n, C = diag{Ci}, and X = diag{Xi}, i ∈{1, 2, …,

n}. In addition, the load voltages are denoted by uL = [un+1

un+2 … un+m]T.

Next, applying the Kirchoff’s and Ohm’s laws, we have

1

, 1,2, ,n m

i ij i j

j

i y u u i n m

(4)

where yij is the cable conductance, yij=0 if there is no cable

connecting nodes i and j, and rij =1/yij represents the resistance.

Writing (4) in block matrix form, we have

1 2

1 2

,

T

S nSS SLS S S

TLS LLL L L L n n n m

i i i iY Yi u uY

Y Yi u u i i i i

(5)

where Y is the symmetric admittance matrix of the graph, YSS

∈ n nR , YSL∈ n mR , YLS ∈ m nR , and YLL∈ m mR are the

corresponding block matrices, iS and iL are the current vectors

of the sources and loads, respectively. For a CPL, it yields

, 1, 2, ,i i iu i P i n n n m (6)

where Pi is the power of load at node i, and the right-hand side

of the equation is negative because the actual current direction

and reference are opposite.

IV. EXISTENCE OF EQUILIBRIUM OF DC MICROGRID

Besides instability, CPLs can cause inexistence of

equilibrium in DC microgrids [19], [20]. For simplicity, all the

loads can be equivalent to a common CPL under the

assumption that the DC-bus resistance can be neglected [18]-

[19]. Consequently, the existence of equilibrium is formulated

by the solvability of a quadratic equation with one unknown,

whose conditions are easily obtained. However, in most cases,

the DC bus resistance cannot be neglected, and the

equilibrium should be determined from quadratic equations

with multiple unknowns. In this case, the topology of the

solution set is complicated [20].

A. Problem Formulation

According to (3), when the system achieves steady-state,

the output voltage is given by uS=V*– KiS, and substituting it

into (5) yields

S SS SS S SL L

L LS LS S LL L

i Y V Y Ki Y u

i Y V Y Ki Y u

, (7)

whose simplification yields to

1 11 1 1

1 11 1 1

S SS SS SS SL L

L LS LS SS LL LS SS SS SL L

i Y K V Y K Y Y u

i Y Y K Y K V Y Y K Y K Y Y u

. (8)

Combining this result with (6), the system admits a constant

steady state if and only if

1

1, 2, ,

L L

i i i

i Y u

u i P i n n n m

(9)

is solvable where

1 1

1 1 1 11= 1 ,ref LS SS SS n LL LS SS SS SLu Y Y Y K Y Y Y K Y K Y Y

.

Clearly, the system admits an equilibrium if and only if, for

given values of V*, Y, and P, the quadratic equations in (9)

admit a real solution.

Let UL = diag{uL}, P = [Pm+1 Pm+2 …Pm+n]T , and f = [ f1

f2 … fm ]T = UL (YLSV*+YLL uL). We define the multidimensional

quadratic mapping, f : Rm→Rm , of the form

2 11 L L m L L Lf f u f u u U β Y uf .

Then, E = {f(uL): uL∈Rm} is the image of the space of

variables uL under this map.

B. Related Results

Some existed studies neglect the resistance of the common

DC bus [18], [19]. Under this assumption, all CPLs in a DC

microgrid are equivalent to a common CPL, as shown in Fig.

2, and (9) becomes

eq LS LL eq equ Y V Y u P (10)

where 1Teq mP P is the equivalent load, ueq∈R is its voltage,

1

11 1TLL n SS nY Y K

and

11=1T

LS n SSY Y K

are the equivalent

admittance matrices. This system admits an equilibrium if and

only if

2

4 0LS LL eqY V Y P (11)

In fact, although (11) holds, system admits no constant

steady-state equilibrium due to the resistance of the DC bus.

In [20], two results based on LMI are presented as:

Proposition 1: Assume there exists a diagonal matrix H =

diag{hi} such that

1 1

11 11

0

2n m T

i ii n

HY Y H

h P H HY Y H H

(12)

Then, (9) has no real solution.

The necessary condition in Proposition 1 implies that if

there exists a constant steady-state, LMI (12) has no solutions.

Proposition 2: If E is convex, (9) have a real solution if and

only if LMI (13) is unfeasible. Reference [21] provides a test

to check convexity of E. However, for m > 2, set E is usually

nonconvex.

eqP

DG4 DG3

DG2DG1 Fig.2. Equivalent topology of DC microgrid with n DGs and one CPL.

In [22], a sufficient condition is obtained by using

contraction mapping, i.e., the quadratic equations are solvable

if

1

14 1diag Y diag

(13)

where 11Y and idiag P . The proof of (13) is

detailed in [22], and here we provide an alternative proof

along with two stronger conditions.

C. Sufficient Conditions for Existence of Equilibrium

By simplifying (9), we have

1L LU Y u P (14)

Multiplying by 1 11 LY U , (14) becomes

1 1 1 11 1 2Θ

T

L L L L n n n mu F u Y g u g u u u u

， (15)

Then, the solvability of (14) is equivalent to the existence of

fixed point in the fractional functions from (15). In addition, if

1) and 2) of Lemma 2 are satisfied, the system admits a

constant steady state.

Now, we state the main results of this paper.

Proposition 3: The following two statements are true

1) Y is irreducible;

2) 11Y is positive ( 1

1Y O );

3) ζ = uref 1m.

Proof. First, we assume that Y is reducible, and as Y is

symmetric, there exists a permutation matrix E such that '

11

'22

TY O

E AEO Y

. (16)

Hence, the DC microgrid can be divided into two separate

microgrids, which contradicts the assumption that all sources

and loads are strongly connected, thus proving 1).

Next, Y1 can be simplified as 1

11 LL LS SS SLY Y Y Y K Y

and we define 1

1ΓSS SL

LS LL

K Y Y

Y Y

. (17)

Clearly, Y1 is a Schur complement of Г1. According to 1) of

Lemma 1, Г1 is a positive-definite M-matrix. Clearly, Г1 is

irreducible as Y, and according to 2) of Lemma 1, 11 is

strictly positive. Applying the formula for the inverse of a

block matrix, we obtain

1 11 1 1 1

11

1 11 1 1 1

1

ΓSS SL LL LS SS SL

LL LS SS SL LL LS

K Y Y Y Y K Y Y Y

Y Y K Y Y Y Y Y

.

Because 11 is strictly positive, 1

1Y is positive and

irreducible, thus proving 2).

Finally, given that Y is a Laplacian matrix, Y1n+m=0n+m, i.e., 1

1

1 1 0

1 1 0

n SS SL m n

LL LS n m n

Y Y

Y Y

. (18)

Then, we have

1 11 1 1

11 1

1 1

1 1 1 1 0

LS LS SS n LL LS SS SS SL m

LS n LL m LS SS n SS SL m m

Y Y K Y K Y Y K Y K Y Y

Y Y Y K Y K Y Y

(19)

According to (18), we obtain 1 11 1 0ref m mu Y , i.e., ζ =

uref 1m, thus proving 3).

Theorem 1. A necessary condition for (15) to be solvable is

2refu , (20)

where χ is the Perron root of 11 ΘY .

Proof. First, let 11 1 2Θ ; ; ; nY a a a where ai is the ith row

vector of 11 ΘY . We assume 0 is a solution of (15) of the

form

1

1, 1,2, ,

m

i ref ij

j j

u a i m

, (21)

where aij represents the vector entry. Thus, (21) can be

expressed as

2 2

1

1, 1,2, ,

2 4

mref ref

i i ij

j j

u ua i m

. (22)

Then, the following can be obtained

2

1

1 10, 1,2, ,

4

mref

ij

ji j

ua i m

, (23)

whose matrix form is given by

2 1 1 1 11 1 24 0 ,

T

ref m nu I Y (24)

According to 2) of Proposition 3, 11Y is positive, i.e.,

2 114refu I Y is a Z-matrix. In addition, (24) satisfies 2) of

Definition 2, and hence 2 1

14refu I Y is an M-matrix.

Therefore, according to 1) in Definition 2, (20) must be

satisfied, completing the proof.

Theorem 2. There must exist a unique vector Lu such that

,L L Lu F u h u provided that

2

1 ,maxref ij

i j mu f q

(25)

where q=[q1 q2 … qm]T is an arbitrary positive vector and

2 1 1 11 2min 4 ,

2

Ti i

ref ref m

i

q a qh u u q q q

q

,

2

4 max , if 2 max ,

if 2 max ,

j j ji i i

i j j i i j

jiij

j i j ji i

j ji i j i i j

j i i j

a q a q a qa q a q a q

q q q q q q

a qa qf q

q q a q a qa q a q

a q a qa q a q q q q q

q q q q

The proof is detailed in Appendix. Remark 1: We transform the quadratic equation solvability

into the existence of a fixed point in a nonlinear mapping. The

key is to construct an appropriate mapping that satisfies the

conditions of Lemma 2. In this process, the positivity of 11Y

play a crucial role. However, for a passive-transmission

network, its admittance matrix Y is a symmetric positive

semidefinite Z-matrix. Thus, Γ1 and Y1 must be irreducible

M-matrices. Therefore, 11Y is positive, and function F(x) is

strictly increasing. This way, we can obtain the sufficient

conditions for the system to admit an equilibrium by using the

Tarski fixed-point theorem.

Furthermore, positive vector q in (25) is arbitrary.

Consequently, we can obtain the optimal q that minimizes the

right side of (25). Thus, the result is less conservative, and the

optimal sufficient conditions can be formulated as

2

1 ,min maxref ij

q i j mu f q

, (26)

Then, to obtain an explicit analytic condition, we take q as

several special vectors into (25).

Corollary 1. For given Y, K and P, the system must admit a

constant steady-state if the following holds

11m 2 Θin ,ref Yu

, (27)

where η is the Perron vector of 11 ΘY , max i and

min i .

Proof. Consider q = 1m in (25), and then

2

1 , 1 ,

11

1

max max 4max 1 , 1

4max 1 4 Θ

ref ij i m j mi j m i j m

i mi m

u f q a a

a Y

. (28)

Substituting ζ = uref 1m into (13), it turns equivalent to (28).

Thus, (13) is proved.

Let q = η, we have

, 2max , 2j j ji i i

j i j i i j

a aa a

. (29)

According to (29), fij(η) becomes

2

2

2

4 max , 4 if

= if

jii j

i j

jiij

j i jii j

j iji

j i

f

, (30)

and hence the following is straightforward

2

max ijf

. (31)

Thus, (27) is obtained, completing the proof.

Remark 2: Corollary 1 shows that for fixed load P, the

system must admit a constant steady-state as long as uref is

large enough, as can be expected. Meanwhile, (26) and (27)

provide numerical and analytical condition to guarantee the

existence of the equilibrium, respectively. Moreover, both (26)

and (27) are less conservative than the results from the recent

works [22].

The relation among line resistances, virtual resistances,

loads and DG output voltages that make (9) solvable is

described by (26) and (27). This allows to determine design

guideline to build reliable DC microgrid.

V. STABILITY ANALYSIS OF DC MICROGRID

A. Small-signal Model around Equilibrium

According to Theorem 2, if condition (26) holds, the system

will admit a constant steady state, we denote it by

, , , ,L S S L Li u i i u . Linearizing (6) around the equilibrium, the

equivalent CPL resistances can be obtained from

2

11, 1, 2, ,i i i i i

i

i u r P u i n n n mr

(32)

where Δ represents the small-signal variation around the

equilibrium and ri is the negative CPL resistance. Substituting

(32) in linearized (5), we have

1

1

L SS SL LL L LS Si Y Y Y R Y u

, (33)

where RL = diag{ri}, and combining the linearized (5) with

(33), the equivalent linearized model of the system is given by

ΔΔ Δ

ΔΔ Δ

LL S

SL eq S

d iX K i u

dt

d uC i Y u

dt

, (34)

where 1

1eq SS SL LL L LSY Y Y Y R Y

. The system Jacobian

matrix is given by 1 1

2 1 1=

eq

X K XJ

C C Y

. (35)

B. Stability Conditions and Stabilization

According to the Hartman–Grobman theorem, the

equilibrium is stable if and only if J2 is Hurwitz. The

characteristic polynomial of J2 is obtained as

1 1 1 1

1 1 1 1

1 1 1 1 1

1 1= +

eq eq

eq

eq

X K X λI X K XλI

C C Y C λI C Y

λI X K λI C Y C λI X K X

λI X K λI C Y I

,(36)

which results in

2 1 1 1 1 1 1 0eq eqλ I λ X K C Y X KC Y X C . (37)

To maintain symmetry in (37), we take X=bK, where b is a

proportionality coefficient. Then, the duty cycle is designed as

1,2, ,i

i ii ref i L i

i i i

L ud u k i u i n

bV k V , (38)

and the system Jacobian matrix is

1

21 1

1 1

=

eq

I Kb bJ

C C Y

. (39)

Then, simplifying (37), we have

2 1 1 1 11 10eq eqλ I λ I C Y C Y K C

b b

(40)

Theorem 3. Matrix J2 is Hurwitz if the following holds

1

0

0

eq

eq

C bY

K bY

(41)

Proof. Multiplying (37) by C, we have

2 11 10eq eqλ C λ C Y Y K

b b

(42)

Then, according to Lemma 3, (41) can be easily obtained.

Combining with the existence condition of equilibrium point,

the system admit a stable operation point if (26) and (41) hold.

Proposition 4: The minimal eigenvalue λ1(Yeq) of Yeq is

negative.

Proof. Clearly, Yeq is the Schur complement of Г2 which is

defined as

2 1

SS SL

LS LL L

Y Y

Y Y R

(43)

Given that1

2 11 1 0

n mTn m n m ii n

r

, Γ2 must have at

least one negative eigenvalue. In fact, 1LR

when L Lu u . According to (22), we obtain 12

refL m

uu , thus,

2 1 1 11 1 14 refu Y Y R O . Then, according to Lemma 3 and (20),

1 1 21 1 4 1refY R u .

Let 1 11 1 LR I Y R , whose eigenvalues clearly have a

positive real part. Thus, we have 1/2 1/2 1/2 1 1/2

2 1 1 1 1 1 0LR Y R Y I Y R Y . (44)

In fact, 1 1/2 1/21 1 2 1LY R Y R Y , and according to (44),

11 0LY R . Since

11

LS SS SLY K Y Y

is positive definite,

according to eigenvalue perturbations theorem, we have

1

1 1 11 0LL L L LS SS SLY R Y R Y K Y Y

, (45)

i.e., 1LLLY R is positive definite. Next, as Γ2 has at least one

negative eigenvalue and 1LLLY R is positive definite,

according to Schur’s theorem [23], Yeq must have at least one

negative eigenvalue, thus completing the proof.

Corollary 2. For given K and uref that satisfy (26) and (27),

the equilibrium is stable if

min

1 1 max

10 min ,

eq eq

Cb

Y Y k

, (46)

where Cmin=min{Ci} and kmax=max{ki}.

Proof. According to the eigenvalue perturbation theorem, (41)

holds if the following is satisfied:

min 1

1max 1

0

0

eq

eq

C b Y

k b Y

(47)

Thus, (46) is obtained, completing the proof.

Overall, the system admits a stable equilibrium when

following these two steps:

Step 1. For given P and Y, selecting the appropriate uref and K

to ensure a constant steady state according to Theorem 2 and

Corollary 1.

Step 2. Selecting the appropriate b to ensure the equilibrium is

stable according to Corollary 2.

VI. CASE STUDY

To verify the presented analyses, we simulate a DC

microgrid with the structure in Fig.1 using MATLAB/

Simulink. The ideal CPL is modeled as a controlled current

source, and the system parameters are listed in Table I.

A. Stabilization Design

According to the proposed stabilization controller, the

converter duty cycles are designed according to (38). Take k1 =

k2 = k3 = k4 =1, then, the duty cycles take the form

5

21,2,3,4

3003 10 i

ii ref L i

ud u i u i

b

where b and uref are the parameters that should be selected to

guarantee that the system admits a stable equilibrium.

B. Voltage Reference to Guarantee the Existence of

Equilibrium

System equilibrium Lu is determined by 1L LU Y u P ,

which can be expressed as 11 ΘL L Lu F u Y g u , where ζ

= uref [1 1 1 1 1 1]T. According to the resistances in Table I, Y1

is given by

1

1.5 1 0 0 0 0

1 11 5 0 5 0

0 5 7.5 2 0 0

0 0 2 2.833 0 0

0 5 0 0 7 2

0 0 0 0 2 2.667

Y

According to Theorem 2 and Corollary 1, provided that

either (26) or (27) holds, there must exist a unique vector

Lu D such that L Lu F u , i.e., the system admits a

constant steady state. Let

11 2 3 4 1

1 ,2 , min max , 2, Θij

q i j mf q Y

We use q* to denote the optimal vector that minimizes

1 ,

max iji j m

f q

. If uref > τ2, the system admits an equilibrium.

Then, to test the correctness of the existence condition of

euilibirium, we evaluated four cases:

Case 1: P=103[1 1 1 0.5 0.5 0.5]T, τ1=89.28, τ2=89.63, τ3=90.6,

τ4 = 92.19, q* = [2.3130 2.3167 2.1302 1.7741 2.2724 1.9308]T,

uref =89.64;

Case 2: P =103[1 1 1 0.5 0.5 0.5]T, uref =89.6;

Case 3: P =103[2 2 2 1.5 1.5 1.5]T, τ1 =134.93, τ2 =135.50, τ3 =

136.67, τ4 =140.39, q*=[1.4284 1.519 1.396 1.1887 1.532

1.3268], uref =135.51;

Case 4: P =103[2 2 2 1.5 1.5 1.5]T, uref =135.4.

The corresponding results, obtained from MATLAB, are listed

in Table II. TABLE I.

Simulated System Parameters

Parameter Symbol(unit) Value

Line resistance rij (Ω) r15=r56=r37=1, r67=r28=r69=0.2,

r78 = r9,10= r4,10=0.5.

Converter input voltage

Vi(V) V1=V2=V3=V4=300

Converter inductances

and capacitance

Li (mH),

Ci (mF)

L1=L2=L3=L4=2,

C1=C2=C3=2, C4=2.5.

TABLE II.

Equilibrium of the Evaluated Cases

Cases The corresponding hξ, ζ such that

h F h F The solution of equation uL=F(uL)

Case 1 ζ=89.64[1 1 1 1 1 1]T, hξ =[43.25

43.18 46.96 56.39 44.03 51.82]T

uL=[43.57 43.49 47.24

56.59 44.33 52.05]T

Case 2 Inexistence Unsolvable

Case 3 ζ=135.51[1 1 1 1 1 1]T; hξ=[70.03

65.85 71.65 84.15 65.29 75.39]T

uL=[70.16 65.99 71.77

84.23 65.43 75.50]T

Case 4 Inexistence Unsolvable

The results shows that equation uL=F(uL) has a solution if

uref > τ2, and it may have no solution otherwise. In the

evaluated cases, τ2 < τ3 < τ4, which shows that solvability

condition (26) and (27) are stronger than the result in [22].

Moreover, the results shows that there is a solution when

uref >135.51 and no solution when uref =135.4. Hence,

condition (26) is less conservative.

C. Performances of the proposed stabilization method

According to Corollary 2, the equilibrium is stable if (46)

holds. Let us define b0 as

min

0

1 1 max

1min ,

eq eq

Cb

Y Y k

Hence, the system equilibrium is stable if b<b0. To verify

the correctness of the existence and stability condition of

euilibirium, we evaluated three cases:

Case 5. P=103[1 1 1 0.5 0.5 0.5]T, uref =200, b0 =0.062,

b=3×10-3.

Case 6. P =103[0.5 0.5 0.5 0.5 0.5 0.5]T for t < 0.05 and P =

103[1 1 1 0.5 0.5 0.5]T for t ≥ 0.05, uref = 89.64, b0 = 2.15×10-3,

b=1×10-3;

Case 7: P as in case 6, uref = 89.6, b0 = 2.15×10-3, b = 3×10-3;

Moreover, CPLs were activated at t=0.001s in all these

cases. In Case 5, ki=0 for t <0.2 and ki=1 onwards, i.e., the

proposed stabilization acted for t ≥ 0.2. In Case 6 and 7, ki=1

throughout the simulation. The simulation results are depicted

in Fig.3–5.

DG1 DG2 DG3

DG4 DG5 DG6

100

150

200

250

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Volt

age(

V)

Time(s) Fig.3. Load voltages for Case 5

40

50

60

70

80

90 DG1 DG2 DG3

DG4 DG5 DG6

Vo

ltag

e(V

)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Time(s) Fig.4. Load voltages for Case 6, uref > τ2

0 0.01 0.02 0.03 0.04 0.050

20

40

60

80

100DG1 DG2 DG3

DG4 DG5 DG6

Volt

age(

V)

Time(s) Fig.5. Load voltages for Case 7, uref <τ2

In Fig. 3, the system is unstable for t < 0.2 and stabilize

after activating the proposed stabilization method, verifying its

effectiveness. In Case 6, uref > τ2 and the system admits a

stable equilibrium when the loads increase to maximal values

as shown in Fig.4. In contrast, in Case 7, uref < τ2 and the load

voltages collapse as shown in Fig.5. These results confirm the

correctness of the sufficient conditions to the existence and

stability of equilibrium presented in this paper.

VII. CONCLUSIONS

We investigate the existence and stability of equilibrium of

in general DC microgrids with multiple CPLs. A stabilization

method is proposed and the sufficient conditions for the

system admitting a stable equilibrium are derived. We

transform the problem of nonlinear equation solvability into

the existence of fixed-point of an increasing mapping and

obtain the sufficient condition based on Tarski fixed-point

theorem. The sufficient condition is less conservative

comparing with the existing results. We adopt the linearized

equivalent model around the equilibrium and obtained the

stability conditions by analyzing the eigenvalue of the

Jacobian matrix. These conditions provide a design guideline

to build reliable DC microgrids. Finally, the simulation results

verify the correctness of the proposed conditions.

Appendix. Proof of Theorem 2.

Proof. According to 2) in Proposition 3, 11Y is strictly

positive, and hence 11 ΘY is also strictly positive. Consequently,

11 2 1 2 1 0mF x F x Y g x g x for every 1 2 0mx x ,

satisfying 1) of Lemma 2. Likewise, the system admits an

equilibrium if 2) of Lemma 2 is also satisfied. Let x2 =ζ and

x1=hξ, where 1 1 11 2

T

mq q q

and h is an undetermined

positive scalar. Given that 11Y is positive, the following

can be obtained:

F . (48)

Then, the quadratic equation in (14) is solvable if

h F h . (49)

Clearly, (49) can be expressed as

1, 1,2, ,ref i

i

hu a k i m

q h , (50)

and (50) is equivalent to

2 21 1 1 1

1 1

2 2

4 42 2

4 42 2

ref ref ref ref

m m m mref ref ref ref

m m

q a q q a qu u h u u

q q

q a q q a qu u h u u

q q

. (51)

Next, let

2 2

1

4 42 2

mi i i i

i i ref ref ref refi ii

q a q q a qu u u u

q q

， ， .

If Ω ≠ (i.e., F and F h h ), according to

Lemma 2, there must exist a unique vector, Lh u ,

such that L Lu F u .

Clearly, Ω is non-empty if and only if

2 2

2 2

4 42 2

4 42 2

j ji iref ref ref ref

i j

j j i iref ref ref ref

j i

q a qq a qu u u u

q q

q a q q a qu u u u

q q

(52)

holds for every i, j∈{1,2,…, m}. For specific i and j, if qi=qj,

(52) is solvable as

2 4max ,ji

refi j

a qa qu

q q

(53)

If qi ≠ qj, (52) can be expressed as

2 2 22 4 4j ji i

ref ref refj i i j

a q a qa q a qu u u

q q q q

(54)

By solving (54), the solution of (52) is given by

2

2

2

4max , 2max ,

2max ,

j j ji i iref

i j j i i j

ji

j i j ji iref

j i i jj ji i

j i i j

a q a q a qa q a q a qu

q q q q q q

a qa q

q q a q a qa q a qu

q q q qa q a qa q a q

q q q q

, (55)

thus, obtaining (25) and completing the proof.

VIII. REFERENCES

[1] Yujie Gu, Wuhua Li, and Xiangning He, “Passivity-Based Control of

DC Microgrid for Self-Disciplined Stabilization,” IEEE Trans. Power System, vol.30, no.5, pp. 2623-2632, Sept. 2015.

[2] T. Morstyn, B. Hredzak, G. D. Demetriades, and V. G. Agelidis,

“Unified Distributed Control for DC Microgrid Operating Modes,” IEEE Trans. Power System, vol.31, no.1, pp. 802-812, Jan. 2016.

[3] Dong Chen, Lie Xu, and James Yu, “Adaptive DC Stabilizer with

Reduced DC Fault Current for Active Distribution Power System Application,” IEEE Trans. Power System, vol.32, no.2, pp. 1430-1439,

Mar. 2017.

[4] Jianzhe Liu, Wei Zhang, and Giorgio Rizzoni, “Robust Stability Analysis of DC Microgrids with Constant Power Loads,” IEEE Trans.

Power System, to be published.

[5] Jianwu Zeng, Zhe Zhang, and Wei Qiao, “An Interconnection and Damping Assignment Passivity-Based Controller for a DC–DC Boost

Converter With a Constant Power Load,” IEEE Trans. Industrial

Electronics, vol.50,no. 4, pp. 2314-2322, July,2014. [6] Xiaoyong Chang, Yongli Li, Xuan Li, and Xiaolong Chen, “An Active

Damping Method Based on a Supercapacitor Energy Storage System to

Overcome the Destabilizing Effect of Instantaneous Constant Power Loads in DC Microgrids,” IEEE Tans. Energy Conversion, to be

published.

[7] Xin Zhang, Qing-Chang Zhong, and Wen-Long Ming, “A Virtual RLC Damper to Stabilize DC/DC Converters Having an LC Input Filter while

Improving the Filter Performance,” IEEE Trans. Power Electron., vol.

31, no. 12, pp. 8017-8023, Dec. 2016. [8] X. Lu, K. Sun, J. M. Guerrero, J. C. Vasquez, L. Huang and J. Wang,

“Stability Enhancement Based on Virtual Impedance for DC Microgrids

With Constant Power Loads,” IEEE Trans. Smart Grid, vol. 6, no. 6, pp. 2770-2783,Nov. 2015.

[9] Hye-Jin Kim, Sang-Woo Kang, Gab-Su Seo, Paul Jang, and Bo-Hyung,

“Large-Signal Stability Analysis of DC Power System With Shunt Active Damper,” IEEE Trans. Industrial Electronics, vol.63, no. 10, pp.

6270-6280, Oct.,2016.

[10] A. M. Rahimi, G. A. Williamson, and A. Emadi, “Loop-Cancellation

Technique: A Novel Nonlinear Feedback to Overcome the Destabilizing

Effect of Constant-Power Loads,” IEEE Trans. Veh. Technol, vol. 59, no.

2, pp. 650-661, Feb. 2010. [11] Y. Zhao, W. Qiao and D. Ha, “A Sliding-Mode Duty-Ratio Controller

for DC/DC Buck Converters With Constant Power Loads,” IEEE trans.

Industry Applications, vol. 50, no. 2, pp. 1448-1458, Mar. 2014. [12] L. Herrera, Wei Zhang and Jin Wang, “Stability Analysis and Controller

Design of DC Microgrids with Constant Power Loads,” IEEE Trans.

Smart Grid, vol. 8, no.2, pp. 881–888, 2017. [13] R. K. Brayton, and J. K. Moser, “A theory of nonlinear Networks I,”

Quarterly of Applied Mathematics, Vol. 22, pp. 1-33, April 1964. [14] Weijing Du, Junming Zhang, Yang Zhang and Zhaoming Qian,

“Stability Criterion for Cascaded System With Constant Power Load,”

IEEE Trans. Power Electron., vol. 28, no. 4, pp. 1843-1851, Apr. 2013. [15] C. J. Sullivan, S. D. Sudhoff, E. L. Zivi, and S. H. Zak, “Methods of

optimal Lyapunov function generation with application to power

electronic converters and systems,” in Proc. IEEE Electric Ship Technol. Symp., 2007, pp. 267–274.

[16] Dena Karimipour and Farzad R. Salmasi, “Stability Analysis of AC

Microgrids With Constant Power Loads Based on Popov’s Absolute Stability Criterion,” IEEE Trans. Circuits and Systems II, vol. 62, no. 7,

pp. 696-700, July, 2015.

[17] Sandeep Anand, and B. G. Fernandes, “Reduced-Order Model and Stability Analysis of Low-Voltage DC Microgrid,” IEEE Trans.

Industrial Electron., vol. 60, no. 11, pp.5040-5049, Nov. 2013.

[18] Andr e Pires N obrega Tahim, Daniel J. Pagano, Eduardo Lenz, and Vinicius Stramosk, “Modeling and Stability Analysis of Islanded DC

Microgrids Under Droop Control,” IEEE Trans. Power Electron., vol.

30, no. 8, pp. 4597-4607, Aug. 2015. [19] Mei Su, Zhangjie Liu, Yao Sun, Hua Han, and Xiaochao Hou. “Stability

Analysis and Stabilization methods of DC Microgrid with Multiple

Parallel-Connected DC-DC Converters loaded by CPLs,” IEEE Trans. Smart Grid, to be published.

[20] Nikita Barabanov, Romeo Ortega, Robert Griñó, Boris Polyak, “On

Existence and Stability of Equilibrium of Linear Time-Invariant Systems

With Constant Power Loads,” IEEE Trans. Circuit and System, vol. 63,

no. 1, pp. 114-121, Jan. 2016.

[21] B. Polyak and E. Gryazina, “Advances in Energy System Optimization,”

in Proc. ISESO-2015, Heidelberg, Germany. [22] J.W. Simpson-Porco, F. Dörfler, F. Bullo, “Voltage collapse in complex

power grids,” Nature Communication, vol. 7, no. 10790, 2016.

[23] Roger A. Horn, Charles R. Johnson, “Topics in Matrix Analysis,” Cambridge, 1986.

[24] C. D. Meyer, Matrix Analysis and Applied Linear Algebra. Society for

Industrial and Applied Mathematics, Philadelphia (2000). [25] R. S. Varga, “Matrix Iterative Analysis,” Prentice-Hall, Englewood

Cliffs, N.J., 1962.

[26] John Kennan, “Uniqueness of positive fixed points for increasing concave functions on Rn: An elementary result,” Review of economic

Dynamic, 4(4): 893-899, 2001.

[27] F. Tisseur, K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev. 43 (2001) 235–286.